Let 1 be a cube root of unity and S be the set of all non-singular matrices of the form [ array ccc 1 & a & b & 1 & c ^2 & & 1 array ] where each of a, b and c is either or ^2 , then the number of distinct matrices in the set S is

Step-by-step solution

Let A= [ array ccc 1 & a & b & 1 & c ^2 & & 1 array ] For non-singular matrix aligned & | A | 0 & | array ccc 1 & a & b & 1 & c ^2 & & 1 array | 0 aligned aligned & 1(1- c)-a ( - ^2 c )+b(0) 0 & 1(1- c)-a (1- c) 0 & (1- c)(1-a ) 0 & c 1 and a 1 & c ^2 and a ^2 [ ^3=1 ] aligned So possible value of a and c is only and b can take values or ^2 . The possible number of distinct matrices =2 .